Group Theory

Reference notes on group theory — the mathematics of symmetry — from the bare axioms of an abstract group through the structure theory of finite groups, representation and character theory, and the theory of Lie groups and Lie algebras that underlies every continuous symmetry in physics.

These pages are companion material to the rest of the Mathematics section and the Physics tree. They stay self-contained and general: physics is the destination (the Lorentz and Poincaré groups, the gauge groups $U(1)$, $SU(2)$, $SU(3)$, Wigner's classification), but the framing here is mathematical. The manifold and tangent-space prerequisites for the Lie material are taken from topology-manifolds.md, and the $\mathrm{Spin}$/Clifford construction is deferred to clifford-algebra.md.

Contents

A. Foundations — abstract group theory

1. Group Axioms and First Examples — the four axioms, order, abelian groups, and the standard menagerie (${\mathbb{Z}}_n$, $S_n$, $D_n$, $G{L}_n$, $Q_8$).
2. Subgroups, Cosets, and Lagrange's Theorem — the subgroup criterion, cosets as a partition, Lagrange's theorem, normal subgroups, and quotient groups.
3. Homomorphisms and the Isomorphism Theorems — kernels and images, the four isomorphism theorems, automorphisms, and the centre.
4. Group Actions: Orbits, Stabilisers, and Counting — the orbit–stabiliser theorem, Cayley's theorem, the class equation, and Burnside's counting lemma.
5. Direct and Semidirect Products — building groups from pieces; the extension problem and split vs. non-split extensions.

B. Structure theory of finite groups

6. Cyclic and Finitely Generated Abelian Groups — the classification of cyclic groups and the fundamental theorem of finitely generated abelian groups.
7. Permutation Groups $S_n$ and $A_n$ — cycle structure, parity, and the simplicity of $A_n$ for $n\ge 5$.
8. The Sylow Theorems — $p$-groups, the three Sylow theorems, and the classification of groups of small order.
9. Composition Series; Solvable and Nilpotent Groups — Jordan–Hölder, the derived series, and the bridge to Galois theory.
10. Simple Groups and the Classification — the building blocks of finite group theory and the CFSG.
11. Presentations and Free Groups — generators and relations, Coxeter groups, and the word problem.

C. Representation and character theory

12. Representations: Maschke and Schur — linear representations, complete reducibility, and Schur's lemma.
13. Character Theory — characters, the orthogonality relations, and worked character tables.
14. Induced Representations and Frobenius Reciprocity — restriction, induction, and the bridge to Wigner's classification.

D. Lie groups and Lie algebras

15. Lie Groups and the Exponential Map — a group that is also a manifold; one-parameter subgroups and the exponential map.
16. Lie Algebras: Brackets and Structure Constants — the tangent algebra, the Jacobi identity, the Killing form, and BCH.
17. The Classical Matrix Groups — the $GL/SL/O/SO/U/SU/Sp/\mathrm{Spin}$ family, with dimensions, compactness, and accidental isomorphisms.
18. Structure and Classification of Semisimple Lie Algebras — root systems, Dynkin diagrams, and the ABCDEFG classification.
19. Representations of Lie Algebras — weights, the highest-weight theorem, Casimirs, and worked $\mathfrak{su}(2)$, $\mathfrak{su}(3)$.

E. Topology and analysis of groups

20. Connectedness, Covers, and the Fundamental Group — the identity component, the universal cover, and projective representations.
21. Compact Groups, Haar Measure, and Peter–Weyl — why compactness controls representation theory.

F. Worked examples and the physics bridge

22. Worked Classical Examples: $SO(2)$, $SU(2)$, $SO(3)$ — the machinery exercised end-to-end on the smallest non-trivial groups.
23. Why This Matters — The Physics Bridge — spacetime and internal symmetries, spontaneous symmetry breaking, and anomalies.

Reading order

The two halves are largely independent. The abstract / finite arc runs $$\text{axioms}\to \text{subgroups-cosets}\to \text{homomorphisms}\to \text{group-actions}\to \text{Sylow}\to \text{series-solvable},$$ with representation theory (reps-basics → character-theory) sitting on top. The continuous / Lie arc runs $$\text{lie-groups}\to \text{lie-algebras}\to \text{matrix-groups}\to \text{lie-classification}\to \text{reps-lie},$$ with topology-covers and compact-groups supplying the analytic backdrop and examples-classical tying both arcs together. The Lie pages reuse the manifold machinery of topology-manifolds.md §2; readers who only need a specific definition can jump in anywhere. The whole folder feeds QFT/preliminaries.md, which assumes the Lie-group material as background for the Lorentz/Poincaré classification.